Game-Theoretic and Measure-Theoretic Expectations

Definitions and mathematical statement

The supplied record asks for a comparison between game-theoretic expectations and supermartingales and their measure-theoretic counterparts, but does not specify the underlying game or probability model.

For clarity, the standard measure-theoretic notions can be defined after choosing a probability space $(\Omega,\mathcal F,P)$. For an integrable real-valued random variable $X$, its expectation is \[ \mathbb E_P[X]=\int_\Omega X\,{\,\mathrm{d}} P. \] Given a linearly ordered time index set $T$ and a filtration $(\mathcal F_t)_{t\in T}$, a real-valued process $(M_t)_{t\in T}$ is a $P$-supermartingale if it is adapted, satisfies $\mathbb E_P[|M_t|]<\infty$ for every $t\in T$, and satisfies \[ \mathbb E_P[M_t\mid\mathcal F_s]\leq M_s \quad P\text{-almost surely},\qquad s,t\in T,\ s\leq t. \] These definitions provide comparison terminology; the source does not prescribe this probability space, time index, or integrability class, and does not specify whether the intended measure-theoretic comparison uses one probability measure or a family of measures.

The original excerpt additionally mentions a class $\mathcal Z$ of admissible gambles and the optional assumption that it is scalable. The displayed text is corrupted, but its stated meaning is closure under nonnegative scaling: \[ Z\in\mathcal Z,\quad c\in{\mathbb{R}},\quad c\geq0 \quad\Longrightarrow\quad cZ\in\mathcal Z. \] The domain and value range of these gambles are unspecified. No closure under addition or other structural property is supplied.

Source question. Determine how the game-theoretic expectations and supermartingales arising from the intended gambling protocol relate to measure-theoretic expectations and supermartingales, identifying the hypotheses under which a correspondence holds and its limitations. The excerpt also asks about a simplification under the additional scalability assumption above, but refers to eliminating $Z$ from an expression that is absent from the record. Consequently, neither that simplification nor a particular equality, inequality, or equivalence between the two frameworks can be stated precisely from the supplied evidence. The game-theoretic definitions, admissible strategies, timing, comparison measures, and relevant payoff classes remain unresolved.

Short English statement

The question asks when game-theoretic expectations and supermartingales correspond to measure-theoretic ones. The supplied excerpt gives an optional scaling condition on gambles, but omits the gambling protocol and a referenced formula, so a precise comparison theorem cannot yet be specified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005865$, title Game-Theoretic and Measure-Theoretic Expectations, source problem code catalogue entry. The record assigns the year 2024 and names no individual proposer.
• Supplied source citation: Game-theoretic Statistical Inference: Optional Sampling, Universal Inference, and Multiple Testing Based on E-values (2024), Oberwolfach Reports, pp. 1339--1386, DOI 10.4171/owr/2024/24. Original OWR report: https://doi.org/10.4171/owr/2024/24. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Frongillo minimax duality": https://arxiv.org/abs/2512.21026.
• Supplied literature source labeled "safe anytime-valid inference": https://doi.org/10.1214/23-STS894.
• Literature assessment dated 2026-08-22: this is supplied dated curation, not independently verified. The supplied status is partially_solved. The supplied assessment says that minimax duality identifies conditions under which game-theoretic upper expectations and supermartingale constructions coincide with measure-theoretic counterparts, while warning that the correspondence is not universal without structural assumptions on the gamble space and timing. The background calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.